Polynomial localization of the 2D-Vertex Reinforced Jump Process

We prove polynomial decay of the mixing field of the Vertex Reinforced Jump Process (VRJP) on ${\mathbb{Z}}^2$ with bounded conductances. Using [17] we deduce that the VRJP on ${\mathbb{Z}}^2$ with any constant conductances is almost surely recurrent. It gives a counterpart of the result of Merkl, Rolles [14] and Sabot, Zeng [17] for the 2-dimensional Edge Reinforced Random Walk.